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Chapter 21: Problem 53

An electric dipole with dipole moment \(\vec{p}\) is in a uniform external electric field \(E\). (a) Find the orientations of the dipole for which the torque on the dipole is zero. (b) Which of the orientations in part (a) is stable, and which is unstable? (Hint: Consider a small rotation away from the equilibrium position and see what happens.) (c) Show that for the stable orientation in part (b), the dipole's own electric field tends to oppose the external field.

Short Answer

Expert verified
For (a), the orientations for which the torque on the dipole is zero are when \(\theta = 0\) and \(\theta = 180^\circ\). For (b), the stable equilibrium is when \(\theta = 0\), while the unstable equilibrium is when \(\theta = 180^\circ\). For (c), at the stable position, the electric field of the dipole tends to oppose the external electric field.

Step by step solution

01

Determine the Orientations for Zero Torque

The torque \(\tau\) on a dipole in an external electric field is given by the equation \(\tau = pE \sin \theta\), where \(\theta\) is the angle between the dipole moment vector and the electric field vector. Torque is zero when \(\sin \theta = 0\), which happens when \(\theta = 0\) (dipole aligns with the electric field direction) or \(\theta = 180^\circ\) (dipole aligns in the opposite direction to the electric field).
02

Determine the Stability and Instability of the Orientations

For the stability determination, consider a small rotation away from the equilibrium positions \(\theta = 0\) and \(\theta = 180^\circ\). Now, after rotating, for \(\theta = 0^\circ\), the torque will tend to rotate the dipole back to its original position, making it a stable equilibrium. On the other hand, after rotating for \(\theta = 180^\circ\), the torque will tend to rotate in the same direction of rotation , making it an unstable equilibrium.
03

Show the Opposition of the Dipole鈥檚 Electric Field to the External Field

Consider the stable orientation, which is when \(\theta = 0^\circ\). The electric field \(E'\) produced by the dipole at large distances is opposite to the orientation of the dipole. So, for this stable condition, the external electric field is aligned with the dipole, and thus the electric field produced by the dipole itself points in the opposite direction. Thus, the field of the dipole opposes the external electric field.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Torque on Electric Dipole
When an electric dipole with a dipole moment \( \vec{p} \) is placed in a uniform electric field \( E \) it experiences a torque \( \tau \). This torque is calculated by the cross-product of the dipole moment and the electric field, leading to the formula \( \tau = pE \sin(\theta) \) where \( \theta \) is the angle between the dipole moment and the electric field vectors.

At the angles \( \theta = 0^\circ \) and \( \theta = 180^\circ \) the sine function is zero, hence the torque is zero, indicating equilibrium positions. Understanding the relationship between torque and the dipole's orientation is crucial for applications ranging from molecular dynamics to the functioning of electronic devices.
Electric Dipole Stability
The stability of an electric dipole in an external field can be analyzed by perturbing it slightly from its equilibrium position. If the dipole at \( \theta = 0^\circ \) is displaced, the resulting torque will act to restore it to its initial position, akin to a marble resting at the bottom of a bowl鈥攊llustrating a stable equilibrium.

Conversely, if the dipole at \( \theta = 180^\circ \) is perturbed, the produced torque will further displace it from equilibrium, resembling a marble on top of a hill鈥攖his is known as an unstable equilibrium. The behavior under small displacements is often examined in physics to ascertain stability in systems ranging from electric dipoles to mechanical constructions.
Dipole's Electric Field Opposition
In a stable configuration, where the dipole moment aligns with the electric field (\( \theta = 0^\circ \) ), any external influence is naturally opposed by the dipole's response. The electric field \( E' \) generated by the dipole opposes the applied uniform electric field \( E \) because they are in opposite directions.

This opposition is comparable to a feedback mechanism where the system acts to minimize the effect of a disturbance, which in this case is the external electric field. Recognizing this opposing nature helps in understanding phenomena such as polarization in materials and the fundamental principles of electromagnetism. This knowledge is not just theoretical but is also applied in designing sensors and other electronic components.

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Most popular questions from this chapter

Two thin rods, each with length \(L\) and total charge \(+Q,\) are parallel and separated by a distance \(a .\) The first rod has one end at the origin and its other end on the positive \(y\) -axis. The second rod has its lower end on the positive \(x\) -axis. (a) Explain why the \(y\) -component of the net force on the second rod vanishes. (b) Determine the \(x\) -component of the differential force \(d F_{2}\) exerted on a small portion of the second rod, with length \(d y_{2}\) and position \(y_{2},\) by the first rod. (This requires integrating over differential portions of the first rod, parameterized by \(\left.d y_{1} .\right)\) (c) Determine the net force \(\vec{F}_{2}\) on the second rod by integrating \(d F_{2 x}\) over the second rod. (d) Show that in the limit \(a \gg L\) the force determined in part (c) becomes \(\frac{1}{4 \pi \epsilon_{0}} \frac{Q^{2}}{a^{2}} \hat{\imath}\). (e) Determine the external work required to move the second rod from very far away to the position \(x=a\), provided the first rod is held fixed at \(x=0 .\) This describes the potential energy of the original configuration. (f) Suppose \(L=50.0 \mathrm{~cm}, a=10.0 \mathrm{~cm}, Q=10.0 \mu \mathrm{C},\) and \(m=500 \mathrm{~g}\). If the two rods are released from the original configuration, they will fly apart and ultimately achieve a particular relative speed. What is that relative speed?

Excess electrons are placed on a small lead sphere with mass \(8.00 \mathrm{~g}\) so that its net charge is \(-3.20 \times 10^{-9} \mathrm{C}\). (a) Find the number of excess electrons on the sphere. (b) How many excess electrons are there per lead atom? The atomic number of lead is \(82,\) and its atomic mass is \(207 \mathrm{~g} / \mathrm{mol} .\)

(a) What must the charge (sign and magnitude) of a \(1.45 \mathrm{~g}\) particle be for it to remain stationary when placed in a downwarddirected electric field of magnitude \(650 \mathrm{~N} / \mathrm{C} ?\) (b) What is the magnitude of an electric field in which the electric force on a proton is equal in magnitude to its weight?

A charge \(+Q\) is located at the origin, and a charge \(+4 Q\) is at distance \(d\) away on the \(x\) -axis. Where should a third charge, \(q\), be placed, and what should be its sign and magnitude, so that all three charges will be in equilibrium?

Two small spheres, each carrying a net positive charge, are separated by \(0.400 \mathrm{~m}\). You have been asked to perform measurements that will allow you to determine the charge on each sphere. You set up a coordinate system with one sphere (charge \(q_{1}\) ) at the origin and the other sphere (charge \(q_{2}\) ) at \(x=+0.400 \mathrm{~m}\). Available to you are a third sphere with net charge \(q_{3}=4.00 \times 10^{-6} \mathrm{C}\) and an apparatus that can accurately measure the location of this sphere and the net force on it. First you place the third sphere on the \(x\) -axis at \(x=0.200 \mathrm{~m} ;\) you measure the net force on it to be \(4.50 \mathrm{~N}\) in the \(+x\) -direction. Then you move the third sphere to \(x=+0.600 \mathrm{~m}\) and measure the net force on it now to be \(3.50 \mathrm{~N}\) in the \(+x\) -direction. (a) Calculate \(q_{1}\) and \(q_{2}\). (b) What is the net force (magnitude and direction) on \(q_{3}\) if it is placed on the \(x\) -axis at \(x=-0.200 \mathrm{~m} ?\) (c) At what value of \(x\) (other than \(x=\pm \infty\) ) could \(q_{3}\) be placed so that the net force on it is zero?
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